Difference between revisions of "Wallis's formula"

 
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'''Wallis's formula''' states that
 
'''Wallis's formula''' states that
  
(1)<math>\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\left(\frac{1}{2}\right)\left(\frac{2}{3}\right)\cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right)=\frac{\pi}{2}\prod_{k=2}^n\left(\frac{k-1}{k}\right)</math> for even n.
+
(1)<math>\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right)</math> for even <math>n\geq2</math>
  
(2)<math>\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\displaystyle\left(\frac{2}{3}\right)\left(\frac{3}{4}\right)\cdots\left(\frac{n-1}{n}\right)=\prod_{k=3}^n\left(\frac{k-1}{k}\right)
+
(2)<math>\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\displaystyle\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\cdots\left(\frac{n-1}{n}\right)</math> for odd <math>n\geq3</math>
</math> for odd n.
 
  
  
Wallis's formula often works well in combination with [[trigonometric substitution]] in reducing complicated definite integrals with more manageable ones.
+
Wallis's formula often works well in combination with [[trigonometric substitution]] in reducing complicated definite integrals to more manageable ones.

Revision as of 03:10, 9 July 2006

Wallis's formula states that

(1)$\displaystyle\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\left(\frac{1}{2}\right)\left(\frac{3}{4}\right)\cdots\left(\frac{n-1}{n}\right)\left(\frac{\pi}{2}\right)$ for even $n\geq2$

(2)$\int_0^{\frac{\pi}{2}} \cos^n(x)\,dx=\displaystyle\left(\frac{2}{3}\right)\left(\frac{4}{5}\right)\cdots\left(\frac{n-1}{n}\right)$ for odd $n\geq3$


Wallis's formula often works well in combination with trigonometric substitution in reducing complicated definite integrals to more manageable ones.